A non-Turing model of computation - IIT Kanpur

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A non-Turing model of computation Nitin Saxena (CSE@IIT Kanpur, India) (*Thanks to the artists for their images. ) July 2020, NPTEL Special Live Series

Transcript of A non-Turing model of computation - IIT Kanpur

A non-Turing model of computation

Nitin Saxena (CSE@IIT Kanpur, India)

(*Thanks to the artists for their images.)

July 2020, NPTEL Special Live Series

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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What's computing?The first response was by Alonzo Church (1935-6).

Using effective computability based on his λ-calculus.

Soon, Alan Turing (1936) postulated a simple, mostgeneral, mathematical model for computing – Turing machine.

Church (1903-1995)

Turing (1912-1954)

100th bday!

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Church & TuringComputation is: whatever can be simulated on a Turing machine (TM).

TM invented to resolve Hilbert (1928)'s question-- “design an algorithm to decide whether a given statement is provable from the axioms using logic”.

The answer requires defining 'algorithm'.hence, 'computation' requires a new mathematical framework.

Algorithm is very much like a program.TM is like a real computer-- highly iterative.

Hilbert (1862-1943)

Entscheidungsproblem

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Church & Turing

∼ ⋀ ⋁

x2

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x1 x

4x

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Φ(x)

TM defines time & space complexity of a problem.Asymptotics: as a function of input size n.

OPEN Qn: Is there a problem that requires 2n time on a TM?

Consider SATisfiability problem. Boolean formulas. You get the famous P≠NP question.

It's unclear how to prove the impossibility of a faster algorithm.

Since math proofs need an algebraic or geometric structure.

So, let us look at an algebraic analogue---

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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Valiant: Algebraic circuitsAn arithmetic circuit Φ has gates {+, ∗}, variables {x

1,....,x

n} and constants from

some field F.Defines size, depth, fanin, fanout.

An arithmetic circuit is an algebraically neat model to capturereal computation.

It's complete for polynomials!

Valiant (1977) formalized computation & resources using algebraic circuits.

Giving birth to his VP≠VNP question!

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Leslie Valiant (1949-)

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Valiant: Algebraic circuitsIn n-variate n-degree polynomial f(x), there are at most n+nC

n

≈ 2n monomials. “Usually”, that's the circuit-size complexity of f(x). What's circuit-depth of f(x) ?

OPEN Qn: Is there explicit polynomial f(x) that requires 2n size algebraic circuits?

Valiant's VP≠VNP question pin-points this f(x).

To better understand this, let's start with a familiar polynomial--- Determinant.

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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Warmup: Determinant

Jacques Binet (1786--1856)

Augustin-Louis Cauchy (1789--1857)

Determinant is an n=m2-variate m-degree polynomial defined as, eg. for m=3,

It has m*(m-1)*...*1 = m! ≈ mm/2 ≈ n√n/4 monomials.

Qn: What's its circuit-size complexity ?As large as the number of monomials??

x12 x21x33 − x13 x22x31 −Det[x11 x12 x13

x21 x22 x23

x31 x32 x33] = x11 x22x33 −

x11 x23x32+x12x23x31+x13 x21 x32 .

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Warmup: DeterminantTheorem [Csanky 1976]: Det

m has a circuit of size < m7.

How's this possible?Algebraic ideas/ identities !

Idea 1: Detm(X) = product of the m eigenvalues of X =: ∏ λ

i .

Idea 2: ∏λi is an expression in p

k := ∑

1≦i≦m λ

ik (1≦k≦m).

Newton's identity.

Idea 3: In turn, pk = tr(Xk) .

Thus, Det(X) computation reduces to matrix-powering Xk.Easy to implement in a circuit.

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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Valiant's permanentValiant (1977) posed a related polynomial– Permanent.

Counts the number of satisfying assignments of a given boolean formula.

Or, number of matchings in a graph.

Its computation solves NP-hard problems!

Valiant's VP≠VNP question: Per

m requires 2m size algebraic circuits?

x12 x21x33+ x13 x22x31+ Per [x11 x12 x13

x21 x22 x23

x31 x32 x33] = x11 x22 x33+

x11 x23x32+x12x23x31+x13 x21 x32 .

OPEN Qn: Given a matrix X compute Perm(X) ?

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Valiant's permanentShow that permanent affords no ultra-clever algebraic identities that help in circuit computation?

Classical algebra is not developed well enough to answer this question.

Permanent, circuits are both recent constructs.A theory specialized to size is missing.

We need to study circuit identities---

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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Zero or nonzeroQuestion: Test whether a given circuit is zero.

Polynomial identity testing (PIT).

PIT has a simple randomized fast algorithm:Evaluate Φ(p) for a random point p.

OPEN Qn: Is PIT in deterministic polynomial time?

Work in last decades show:

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Φ(x)

Meta-Theorem: A solution of identity testing would answer the permanent question.

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Contents

Church & Turing

Valiant: Algebraic circuits

Warmup: Determinant

Valiant's question

Zero or nonzero

Applications

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Application 1-- Graph theoryIs there a perfect matching in a given graph?

First solution by Dinitz (1970), Edmonds, Karp (1972).Deterministic polynomial time using flows.

Is there a fast parallel algorithm?

Relates to PIT.Write it as a determinant, circuit....

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Application 2-- Number theoryCircuits can compute high-degree polynomials.

Eg. f(x) = x2^s requires only circuit-size s.Repeated squaring.

Testing primality of n reduces to testing---

Used by Agrawal-Kayal-S. (2002) in primality testing.First deterministic polynomial time primality test.

Relates to derandomizing PIT.Fix x to special values!

(x+1)n = xn+1 mod n .

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Application 3-- Learning theoryAreas like Artificial Intelligence / Machine Learning model decision-making using circuits.

Artificial Neural Networks (ANN).

ANN is a specialized algebraic circuit.

Backpropagation methods are used to modify the edges, and their weights, to improve the output.

Relates to better understanding of algebraic circuits?Circuit complexity of problems...

Thank you!